PHYSICAL REVIEW E 74, 011120 共2006兲
Nonanalytic microscopic phase transitions and temperature oscillations in the microcanonical
ensemble: An exactly solvable one-dimensional model for evaporation
Stefan Hilbert*
Institute for Physics, Humboldt-Universität zu Berlin, Newton-Strasse 15, D-12489 Berlin, Germany
Jörn Dunkel†
Institute for Physics, Universität Augsburg, Universitätsstrasse 1, D-86135 Augsburg, Germany
共Received 11 October 2005; revised manuscript received 28 February 2006; published 26 July 2006兲
We calculate exactly both the microcanonical and canonical thermodynamic functions 共TDFs兲 for a onedimensional model system with piecewise constant Lennard-Jones type pair interactions. In the case of an
isolated N-particle system, the microcanonical TDFs exhibit 共N − 1兲 singular 共nonanalytic兲 microscopic phase
transitions of the formal order N / 2, separating N energetically different evaporation 共dissociation兲 states. In a
suitably designed evaporation experiment, these types of phase transitions should manifest themselves in the
form of pressure and temperature oscillations, indicating cooling by evaporation. In the presence of a heat bath
共thermostat兲, such oscillations are absent, but the canonical heat capacity shows a characteristic peak, indicating the temperature-induced dissociation of the one-dimensional chain. The distribution of complex zeros of
the canonical partition may be used to identify different degrees of dissociation in the canonical ensemble.
DOI: 10.1103/PhysRevE.74.011120
PACS number共s兲: 05.70.Ce, 05.70.Fh, 36.40.Ei, 64.60.Cn
I. INTRODUCTION
Many macroscopic systems exhibit sudden variations of
their physical properties 共elasticity, conductivity, etc.兲, when
one or more external control parameters 共e.g., energy E, temperature T兲 pass certain critical values 关1,2兴. Such phenomena are usually referred to as phase transitions 共PTs兲. Seminal contributions to the theory of PTs are due to, e.g., Mayer
et al. 关3–5兴, Onsager 关6兴, van Hove 关7兴, Yang and Lee 关8,9兴,
Fisher 关10兴, Grossmann et al. 关11–13兴, Burkhardt 关14兴, Pettini et al. 关15–17兴, and Cuesta and Sánchez 关18兴. These authors have studied in detail canonical ensembles 共CEs兲 and
grandcanonical ensembles, corresponding to systems in contact with heat bath and particle reservoirs.
Singular, or equivalently nonanalytic, PTs are indicated
by a discontinuity in the thermodynamic functions 共TDFs兲 or
one of their derivatives 关19兴. Sometimes it is also useful to
consider smooth PTs, characterized by a strong but analytic
variation in the TDFs 关20–23兴. In the presence of a heat bath,
singular canonical PTs can occur in the thermodynamic limit
only 关8–13兴, whereas finite canonical systems may, at best,
exhibit smooth PTs 关20–23兴. However, the situation changes
if the system under consideration is thermally isolated, corresponding to a microcanonical ensemble 共MCE兲. Due to the
different physical conditions underlying MCE and CE, respectively, one can obtain significantly different predictions
for several observable quantities 关24–29兴. For example, in
certain cases, microcanonical heat capacities can also be
negative 共e.g., in self-gravitating systems兲 whereas canonical
heat capacities are generally positive. In particular, as will
also be shown below, the microcanonical TDFs of finite isolated systems may exhibit nonanalyticities. These singularities reflect evaporation/dissociation phenomena and may be
interpreted as microscopic PTs in the small system 关30兴.
The main objective of this paper is to exemplify the differences between the MCE and CE and to elucidate particular observational consequences by means of a onedimensional model for evaporation. Remarkably, this model
system analyzed below allows for calculating exactly both
the canonical and microcanonical TDFs for an arbitrary number of particles. The knowledge of the exact TDFs for both
ensembles provides the basis for a detailed comparison of
observables. Our main results can be summarized as follows.
In the case of the MCE the model system exhibits 共N
− 1兲 singular microscopic PTs, reflected by nonanalytic kinks
in the caloric curve T共E兲 and the pressure curve P共E兲 at
certain critical energy values Ek. The values Ek can be identified with the binding energy of different dissociation states;
i.e., the singularities 共nonanalyticities兲 separate energetically
different evaporation phases. These microscopic PTs are accompanied by strong temperature oscillations; i.e., the temperature of the system decreases when increasing the energy
in the vicinity of the critical values Ek. This effect corresponds to cooling by evaporation 共or dissociation兲. By contrast, nonanalytic transitions are absent in the corresponding
CE; i.e., if the system is embedded into a heat bath of temperature T. Nevertheless, a smooth PT is observed that also
persists in the thermodynamic limit—even though the existence of a singular macroscopic PT is excluded by the 共generalized兲 van Hove theorem 关7,18兴. Finally, our study of the
distribution of complex zeros 共DOZ兲 of the canonical partition function 关20–23兴 suggests that the DOZ may be used to
identify different degrees of dissociation in the CE.
II. THE MODEL
*Electronic address: hilbert@mpa-garching.mpg.de
†
Electronic address: joern.dunkel@physik.uni-augsburg.de
1539-3755/2006/74共1兲/011120共7兲
We consider a one-dimensional model system corresponding to N identical point particles confined by a onedimensional box of size L. The Hamiltonian reads
011120-1
©2006 The American Physical Society
PHYSICAL REVIEW E 74, 011120 共2006兲
STEFAN HILBERT AND JÖRN DUNKEL
H共p,q;L,N兲 =
p2
+ U共q;L,N兲 = E,
2m
共1兲
with q = 共q1 , . . . , qN兲 denoting the coordinates and p
= 共p1 , . . . , pN兲 the conjugate momenta. In the case of an isolated system the total energy E is conserved. The potential
energy U = Uint + Ubox is determined by the interaction potential
C共N兲 ⬅
1
兺 Upair共兩qi − q j兩兲
2 i,j=1
N−1
⍀ = C 兺 ␻k共E + kU0兲N/2⌰共E + kU0兲,
共2a兲
where, for L ⬎ 共N − 1兲共r0 + dhc兲,
Ubox共q;L,N兲 =
再
␻k共N,L兲 =
0,
q 苸 关0,L兴N ,
+ ⬁,
otherwise.
共2b兲
The pair potential is given by
冦
⬁,
r ⱕ dhc ,
Upair共r兲 = − U0 ,
0,
dhc ⬍ r ⬍ dhc + r0 ,
共2c兲
冉 冊兺 冉 冊
k
0 ⬍ r0 ⱕ dhc .
The latter condition ensures that particles may interact with
their nearest neighbors only. Furthermore, we assume that
L ⬎ Lmin ⬅ 共N − 1兲共dhc + r0兲, i.e., the volume is assumed to be
sufficiently large for realizing the completely dissociated
state, corresponding to U = 0. The energy E of the system can
take values between the ground state energy
E0 ⬅ − 共N − 1兲U0
k BT =
␻k共E + kU0兲N/2⌰共E + kU0兲
兺
k=0
2
N N−1
␻k共E + kU0兲
兺
k=0
k BT
=
2
共3a兲
where kB is the Boltzmann constant and
dq
RN
dp⌰共E − H兲
RN
共3b兲
the phase volume 关h is Planck’s constant, and ⌰共x兲 ⬅ 0 for
x ⬍ 0 and ⌰共x兲 ⬅ 1 for x ⱖ 0兴. Using N-dimensional spherical
momentum coordinates, one can rewrite Eq. 共3b兲 as
⍀ = C共N兲
冕
RN
dq共E − U兲N/2⌰共E − U兲,
共4a兲
,
共6兲
⌰共E + kU0兲
冓 冔
p2i
,
2m
i = 1, . . . ,N,
共7兲
where 具·典 denotes the average with respect to the microcanonical probability density function
III. MICROCANONICAL ENSEMBLE
The microcanonical ensemble 共MCE兲 refers to an isolated
system. Thence the control parameters are energy E, volume
L, and particle number N. The thermodynamic 共Hertz兲 entropy of the MCE is given by 关30–33兴
N/2−1
which reduces to the ideal gas law E = NkBT / 2 in the limit
E NU0. It is worthwhile to recall that, for a Hamiltonian of
the form 共1兲, the thermal energy 共6兲 derived from the Hertz
entropy is directly related to the microcanonical mean kinetic
energy per degree of freedom by virtue of the equipartition
theorem 关33,34兴:
f共q,p兲 =
冕 冕
i=0
Given Eqs. 共5兲, the microcanonical temperature T and pressure P are obtained from the standard definitions T−1
⬅ ⳵S / ⳵E and P / T ⬅ ⳵S / ⳵L 关2,33–35兴. For example, for the
temperature one finds
and infinity.
1
N ! hN
k
共− 1兲i
i
N−1
where dhc ⬎ 0 is the hard-core diameter of a particle with
respect to pair interactions. The interaction potential 共2c兲 can
be viewed as a simplified Lennard-Jones potential. The depth
of the potential well is determined by the binding energy
parameter U0 ⬎ 0 and the interaction range by the parameter
r0, where we shall additionally impose that
⍀共E,L,N兲 =
k
N−1
⫻关L − 共N − 1兲dhc − r0共N − 1 − k + i兲兴N . 共5b兲
r ⱖ dhc + r0 ,
S共E,L,N兲 = kB ln ⍀共E,L,N兲,
共5a兲
k=0
i⫽j
and the box potential
共4b兲
where ⌫ denotes the Euler gamma function. For Hamiltonian
共1兲 one can calculate integral 共4a兲 exactly, yielding 共see the
Appendix兲
N
Uint共q;N兲 =
2共2␲m兲N/2
,
⌫共N/2兲N ! NhN
冉 冊
⳵⍀
⳵E
−1
1
␦关E − H共q,p兲兴.
N ! hN
共8兲
Hence for isolated ergodic systems with an arbitrary particle
number N, the caloric law T共E兲 can be measured experimentally by monitoring the kinetic energy over a sufficiently
long time interval 共at fixed energy values E兲.
As shown in Fig. 1, the microcanonical caloric law 共6兲 as
well as the pressure P共E兲 exhibit N nonanalytic points at the
energies Ek = −kU0, k = 0 , . . . , N − 1, separating N energetically different dissociation states 共all bindings intact, one
binding broken, etc.兲. The formal order 关30,36,37兴 of these
nonanalyticities equals N / 2, i.e., the entropy has continuous
derivatives up to order 共N / 2 − 1兲, but the 共N / 2兲th derivative
becomes discontinuous 共a similar result was obtained recently by Kastner and Schnetz for the mean-field spherical
spin model 关38兴; see also Gross 关39兴 for a general discussion
of differentiability properties of the microcanonical partition
function兲. Consequently, the “microscopic 共dissociation兲
phases” as well as the singularities appear to be smoothened
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NONANALYTIC MICROSCOPIC PHASE TRANSITIONS¼
FIG. 1. 共a兲 Microcanonical temperature T and 共b兲 pressure P as
a function of energy per particle ␯ = E / N for a 共reduced兲 density n
= N / 关L − 共N − 1兲dhc兴 = 0.001/ r0 and different number of particles N
= 5 共dashed line兲, N = 15 共dotted兲, and N = 500 共solid兲. Note that each
of the curves is 共N / 2 − 2兲 times differentiable.
out in the thermodynamic limit N → ⬁. Nevertheless, for finite systems—and in particular at small densities—the
nonanalytic behavior is accompanied by strong variations/
oscillations of observable quantities as temperature and pressure, when continuously varying E. Both qualitatively and
quantitatively, this behavior is analogous to what is usually
denoted as a “phase transition.” However, since these microscopic nonanalyticities do not survive in the thermodynamic
limit 共at least for our one-dimensional model兲, they strictly
speaking are not covered by the conventional definition of
singular macroscopic PTs. We shall, therefore, speak of singular (or nonanalytic) microscopic PTs in the MCE 关48兴.
Let us also briefly discuss the parameter dependence of
the microcanonical TDFs shown in Fig. 1. As evident from
Eq. 共5a兲, the positions Ek of the singular macroscopic PTs are
just proportional to U0. The amplitude of the associated oscillations in T and P does also depend on the particle number
N and box size L: The strength of the oscillations increases
for larger values U0 and L, but becomes smaller for larger
particle numbers N. The number and formal order of the PTs,
however, only depend on the particle number N and are independent of U0 and L 共as long as U0 ⬎ 0 and L ⬎ Lmin兲.
Thus, qualitatively, the results are independent of the particular choice of the model parameters U0 and L. Moreover,
analogous features can be found in the microcanonical caloric curves of one-dimensional 共1D兲 Lennard-Jones chains
关30兴.
It should be mentioned that the exact phase volume 共5a兲
of our model system resembles in structure the phase volume
obtained by the harmonic superposition method 共HSM兲 applied to Lennard-Jones clusters 共see, e.g., Doye 关40兴 or Wales
and Doye 关25兴 and references therein兲. The HSM approxi-
mates the phase volume ⍀共E兲 by a sum of ellipsoidal regions
around all local minima of the potential U lower than the
total energy E. This method has been successfully applied to
describe melting phenomena, as, e.g., the low-temperature
properties of three-dimensional Lennard-Jones clusters and
their transition from a solidlike state, where the cluster only
vibrates around the ground state configuration, to a liquidlike
state, where also other locally stable configurations are energetically accessible. The standard HSM approximation, however, does not properly account for the contribution to the
phase volume stemming from 共partly兲 dissociated 共or gas兲
states of the cluster, and therefore, is not suitable for describing evaporation phenomena. In particular, the HSM does not
yield any singular microscopic PTs for 1D Lennard-Jones
chains 共where only one locally stable configuration, i.e., the
ground state, exists兲 therewith contradicting exact analytical
and numerical results 关30兴. By contrast, the model system
discussed here—if considered as an approximation to
Lennard-Jones chains—does reproduce these microscopic
PTs related to evaporation 共but, of course, our model cannot
be applied to melting processes because it is onedimensional兲.
It is worthwhile to discuss the microscopic PTs and the
origin of the associated temperature oscillations, as observed
in our model, from a more general point of view. Mathematically, microscopic PTs of the above type arise whenever the
phase volume ⍀ grows nonsmoothly in the vicinity of some
critical energy value E = Ek. This can best be illustrated by
considering the energetically admissible subset of the configuration space
A共E兲 = 兵q 苸 RN兩⌰关E − U共q兲兴 = 1其.
共9兲
The set A共E兲 consists of all position space points q
= 共q1 , . . . , qN兲 that can be occupied by the system at the given
energy value E. Clearly, the boundary of A, denoted by ⳵A,
determines the effective range of the integral in Eq. 共4a兲.
Hence whenever A or ⳵A, respectively, change their shape
in an irregular 共nonanalytic兲 manner, a nonanalyticity in the
phase volume ⍀ may arise 共and, hence, in the TDFs兲. For
example, such an irregular change in the shape of A occurs
when the energy for the next dissociation step is crossed,
since then some parts of the boundary ⳵A suddenly become
determined by the box potential.
It remains to be discussed how the temperature
oscillations—i.e., the regions with negative heat capacity
共also known as “S-bends” or van der Waals-type loops
关49兴兲—arise: In the vicinity of the dissociation energy Ek, the
set A and, thus, also ⍀ and S grow very rapidly, thereby
giving rise to a drop-off in temperature. Geometrically, this
can be viewed as a sudden increase of the “effective dimensionality” of A. Here, “effective dimensionality” refers to the
number of orthogonal configuration space directions in
which A has an extent comparable to the system size L.
Hence, typically, the temperature oscillations appear more
pronounced for larger values of L. From the physical point of
view, the temperature decrease after the kth dissociation step
just means that for energy values slightly larger than Ek the
dissociated fragments have very little kinetic energy 共since
most of the energy has already been used to break the bind-
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PHYSICAL REVIEW E 74, 011120 共2006兲
STEFAN HILBERT AND JÖRN DUNKEL
ing兲. The larger the system the less likely it is that the fragments temporarily recombine in a state of high kinetic energy; i.e., from a probabilistic standpoint, the average 关Eq.
共7兲兴 is then dominated by phase space regions of low kinetic
energy. With regard to practical applications, this means that
one could cool such a small isolated system of bound particles by injecting energy until the fragmentation process sets
in 共cooling by evaporation/dissociation兲.
The above-described features of microscopic PTs are generic and shared by all physical systems that exhibit dissociation and evaporation 共e.g., similar microscopic PTs and
temperature oscillations are also found for small 1DLennard-Jones molecules 关30兴兲. In particular, microscopic
PTs should become more pronounced in two or three dimensions 共since then effective dimension of A grows even more
rapidly at the dissociation levels兲 and also be observable in
quantum systems. By virtue of dissociation experiments with
small particle numbers and very low densities 共similar to
those of Schmidt et al. 关41兴, but without heat bath兲, one
should, in principle, be able to detect the oscillating behavior, e.g., in temperature and pressure curves. However, to
actually observe such oscillations one has to realize the requirements of the MCE, i.e., a thermally isolated system with
regulated energy injection. Furthermore, due to the microscopic origin of the oscillations and the requirement of a
relatively low particle density, a high sensitivity of the velocity and force measurements and long measuring time
spans will be necessary.
IV. CANONICAL ENSEMBLE
Employing the canonical ensemble 共CE兲 is appropriate if
the system under consideration is in thermal contact with a
much larger system 共heat bath兲, as, e.g., realized in dusty
cluster experiments 关42兴. The relevant thermodynamic potential is the free energy 关2兴
F共␤,L,N兲 ⬅ − ␤−1 ln ZC共␤,L,N兲,
共10兲
where ZC is the canonical partition function. The external
control variables are now the inverse temperature ␤
⬅ 共kBT兲−1 of the heat bath, the volume L, and the particle
number N. For the above model, ZC can be exactly calculated, analogous to Eq. 共5兲, as
冉 冊
1 2␲m
ZC =
N! ␤h2
N/2 N−1
␻ke␤kU
兺
k=0
0
共11兲
with ␻k共N , L兲 given by Eq. 共5b兲. Mean energy and pressure
of the CE are defined by Ē ⬅ −⳵共ln ZC兲 / ⳵␤ and P̄
⬅ −⳵F / ⳵L, yielding, e.g., the canonical caloric law
兺k=0 ␻ke␤kU kU0
.
N−1
兺k=0 ␻ke␤kU
N−1
N
−
Ē =
2␤
0
共12兲
0
Figures 2共a兲 and 2共b兲 show Ē共T兲 and P̄共T兲 for different
values of the reduced particle density n = N / 关L − 共N − 1兲dhc兴.
In contrast to the microcanonical pressure 关Fig. 1共b兲兴, the
FIG. 2. 共a兲 Canonical mean energy per particle ¯␯ = Ē / N, 共b兲
pressure P̄, and 共c兲 specific heat capacity c̄L = ⳵Ē / ⳵T 共logarithmic
scale兲 as a function of temperature T for N = 15 particles and different values of the reduced density n = 10−1 / r0 共dashed line兲, n
= 10−3 / r0 共dotted兲, and n = 10−6 / r0 共full兲.
canonical pressure is a monotonous function of T or Ē, respectively. In the thermodynamic limit, microcanonical and
canonical caloric curves become nearly indistinguishable.
The canonical heat capacity c̄L = ⳵Ē / ⳵T exhibits a strong
共nonsingular兲 peak in the temperature region, where dissociation occurs 关Fig. 2共c兲兴. If observed in an experimentally
measured curve, such behavior would possibly be interpreted
as a PT. For decreasing density n, the position of the maximum of c̄L moves closer to T = 0, while its height increases
rapidly. Furthermore, our results indicate that for N ⱖ 15 the
TDFs ¯␯ and c̄L become virtually independent of N. The 共nonsingular兲 peak in the heat capacity persists in the thermodynamic limit 共analogous to the 1D Ising model 关43兴兲.
To obtain a more detailed characterization of the dissociation process in the CE, we next study the distribution of
complex zeros 共DOZ兲 of ZC. As evident from Eq. 共11兲, the
only relevant configurational part of ZC共␤兲 is a polynomial
of degree 共N − 1兲 in z = e␤U0, and, therefore, has 共N − 1兲 complex Fisher zeros 关10兴 per branch of the logarithm. This quasipolynomial structure is a consequence of the fact that, for
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FIG. 3. DOZ of the canonical partition function ZC共␤兲 for N
= 5 共open symbols兲 and N = 15 共closed symbols兲 and n = 10−1 / r0
共triangles兲 and n = 10−6 / r0 共diamonds兲. For better visibility, for N
= 5 共15兲 only the first few branches of zeros with positive 共negative兲
imaginary part are shown 共further branches can be obtained by
shifting with multiples of 2␲i in the vertical direction; generally,
the Fisher zeros are symmetric with respect to the real axis 关20兴兲.
our specific model, the configuration space 关0 , L兴N can be
partitioned into regions of equal total binding energy Ek =
−kU0 , k = 0 , . . . , N − 1 共see Appendix 兲. Since all zeros can be
obtained by adding integer multiples of 2␲i to the zeros ␤k
of the main branch 关for which Im共␤k兲 = ␲兴, it suffices to discuss the main branch only while bearing in mind that each
zero ␤k in the main branch is associated with an infinite set
of zeros 兵␤k + 2␲is 兩 s 苸 Z其. Ordering the zeros according to
their real parts, Re共␤0兲 ⱕ ¯ ⱕ Re共␤N−1兲, we find that the region of the c̄L-peak is well-described by the temperature interval 关Re共␤N−1兲−1 , Re共␤0兲−1兴.
The asymptotic behavior of the DOZ for N → ⬁ may be
used to characterize the parameters 共critical temperature, order, etc.兲 of singular macroscopic PTs 关8–13兴. In our model,
the Fisher zeros are located at least a distance ␲ away from
the real ␤-axis regardless of particle number N 共see Fig. 3兲.
Hence the zeros cannot converge to a point on the real
␤-axis. The peculiar position of the zeros thus ensures agreement with the 共generalized兲 van Hove theorem 关7,18兴 which
excludes the existence of a singular macroscopic PT in our
model.
The DOZ has also been employed to study their finite size
analogs of macroscopic PTs 关20–23兴. Applying these methods to our model, one may interpret the smooth phase transition observed in the CE as a superposition of 共N − 1兲
smooth microscopic “first-order” phase transitions indicated
by the 共N − 1兲 sets of zeros 兵␤k + 2␲is 兩 s 苸 Z其. One can then
use 兵Re共␤0兲 , . . . , Re共␤N−1兲其 to distinguish 共define兲 different
dissociation states in the CE. Thus our results suggest that
the DOZ encodes detailed information about the observed
smooth phase transition and the energetically different degrees of dissociation even if there is no singular PT in the
thermodynamic limit. In particular, since microcanonical
phase volume and canonical partition function can be
mapped onto each other via the Laplace transformation
关44,45兴, one may speculate that there exists a direct mathematical link between the singular microscopic PTs in the
MCE and the DOZ in the CE.
hard-core repulsive part and piecewise constant short-range
attraction. By analyzing the exact TDFs of this model, it was
shown that, in the case of a thermally isolated system
共MCE兲, the microcanonical caloric and pressure laws exhibit
singularities, separating different dissociation states. The formal order of these nonanalyticities increases as the particle
number increases. Hence the microscopic PTs vanish in the
thermodynamic limit and are intrinsically different from the
topologically induced macroscopic PTs discussed in Refs.
关15–17,46,47兴.
For sufficiently low particle numbers and densities, the
microscopic PTs are accompanied by strong oscillations of
the temperature 共mean kinetic energy兲 and pressure. These
oscillations arise from a rapid change of the phase volume
near the dissociation thresholds. They are a generic feature of
particle systems with Lennard-Jones-like interaction potentials 关30,39兴. They are not restricted to one space dimension,
but expected to be even stronger in two and three space
dimensions, and should also be observable in quantum systems. In a suitably designed dissociation experiment, one
should therefore, in principle, be able to detect the oscillating
behavior, e.g., in temperature and pressure curves. In particular, such temperature oscillations may provide the possibility
to cool a small isolated system by means of regulated energy
injection 共cooling by evaporation兲.
If the model system is coupled to a heat bath 共CE兲, a
smooth PT is observed, but no singularities are found 关7,18兴.
Nevertheless, the DOZ seemingly permits one to quantify the
temperature range of the PT and to define different dissociation states.
We thus conclude this paper with two important questions, which need to be answered in the future: Is it possible
to design an experiment that allows one to observe singular
microscopic phase transitions in finite size systems 共or, at
least, oscillatory behavior of thermodynamic observables
such as pressure兲, as predicted by the microcanonical statistical theory? Can one find a direct mathematical link between
singularities in microcanonical partition function of finite
systems and the DOZ of the corresponding canonical partition function—and, thus, between microscopic and macroscopic phase transitions in arbitrary space dimensions?
ACKNOWLEDGMENT
The authors would like to thank M. Kastner and L.
Velazquez-Abad for several helpful remarks.
APPENDIX
To calculate integral 共4a兲, we first eliminate the hard-core
part of the interaction potential by virtue of the transformation L 哫 ␭ = L − 共N − 1兲dhc, qi 哫 xi = qi − nidhc 苸 关0 , ␭兴, here ni
equals the number of particles j with q j ⬍ qi. With these definitions the potential energy can be rewritten as
U⬘共x;␭,N兲 =
V. SUMMARY
We have studied a simple 1D-evaporation model with
nearest-neighbor interaction potentials characterized by a
1
兺 ⬘U⬘ 共兩xi − x j兩兲 + Ubox共x;␭,N兲,
2 i,j pair
where the sum ⌺⬘ goes over nearest neighbors only, and
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STEFAN HILBERT AND JÖRN DUNKEL
⬘ 共r兲 =
Upair
再
− U0 ,
0 ⬍ r ⬍ r0 ,
0,
r ⱖ r0 .
vol共G++
k 兲=
For k = 0 , . . . , N − 1 the family of sets Gk = 兵x 苸 关0 , ␭兴N 兩 U⬘共x兲
= −kU0其 constitutes a partition 共disjoint cover兲 of the configuration space permitted by the transformed box volume
关0 , ␭兴. Thus we can rewrite Eq. 共4a兲 in the form
⫻
␻k ⬅ vol共Gk兲 = N ! · vol共G+k 兲,
where 共writing 兵x 苸 关0 , ␭兴 兩 ¯ 其 = 兵¯其 from now on兲
N−1
k
xk+2−r0
xk+3−r0
dxk+2
r0
dxk+1
0
关xk+1 − r0兴+
vol共G++
k 兲=
dxk ¯
dx关x兴+n =
冕
x3
关x3 − r0兴+
关x兴+n+1
,
n+1
␭
共N−1−k兲r0
⫻¯
· vol共G++
k 兲,
where
where
G++
k
冕
冕
dxN−1
dx2
冕
x2
关x2 − r0兴+
dx1 .
n 苸 N;
共A1a兲
c ⬎ 0.
共A1b兲
By virtue of these identities, we can write Eqs. 共A1兲 as
= 兵x1 ⬍ x2 ⬍ ¯ ⬍ xN Ù U共q兲 = − kU0其.
冉 冊
共N−2−k兲r0
†关x − c兴+ − c‡+ = 关x − 2c兴+,
N
vol共G+k 兲 =
冕
xN−r0
xk+1
冕
In order to calculate vol共Gk兲 we first note that
Next we note that
冕
冕
dxN
The positive part 关x兴+ ⬅ max兵0 , x其 satisfies
k=0
G+k
共N−1−k兲r0
⫻¯
N−1
⍀ = C共N兲 兺 共E + kU0兲N/2⌰共E + kU0兲vol共Gk兲.
冕
␭
k
冕
xN−r0
共N−2−k兲r0
xk+3−r0
dxk+2
r0
冕
dxN−1
xk+2−r0
dxk+1Kk ,
0
冉冊
k
1
关xk+1 − ir0兴+k .
Kk = 兺 共− 1兲i
i
k! i=0
= 兵x1 ⬍ x2 ⬍ ¯ ⬍ xN其 艚 兵共x2 − x1 ⬍ r0兲 Ù ¯ Ù 共xk+1
− xk ⬍ r0兲 Ù 共xk+2 − xk+1 ⱖ r0兲 Ù ¯ Ù 共xN − xN−1
冕
dxN
Performing the remaining integrations and reversing the
transformation L 哫 ␭, one obtains the final result 共5b兲.
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PHYSICAL REVIEW E 74, 011120 共2006兲
NONANALYTIC MICROSCOPIC PHASE TRANSITIONS¼
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